Brainteaser iconBrainteaser - Rabbit Hops

Rabbit Hops
DRW, Citadel
A rabbit starting at position \(0\) on an integer line decides to hop along the integers. At each hop, the rabbit can choose to move forward either \(1\) or \(2\) steps, and it makes this choice with equal probability. Define \( p_k \) as the probability that the rabbit will land exactly on the positive integer position \( k \). What is the largest value of \( p_k \) for \( k > 0 \), and how is it determined?